1. The document discusses the probability that an element of a metacyclic 2-group of positive type fixes a set. It defines metacyclic groups and the concept of a group acting on a set. 2. Previous work on the commutativity degree and the probability that a group element fixes a set is summarized. It was shown that this probability equals the number of conjugacy classes divided by the total number of sets. 3. The main results compute the probability that an element of a metacyclic 2-group of positive type fixes a set, by considering the action of the group on the set of commuting element pairs through conjugation.

1. 71:1 (2014) 7–10 | www.jurnalteknologi.utm.my | eISSN 2180–3722 |
Full paper
Jurnal
Teknologi
The Probability that an Element of Metacyclic 2-Groups of Positive Type
Fixes a Set
Mustafa Anis El-sanfaz
a,
Nor Haniza Sarmin
a*
, Sanaa Mohamed Saleh Omer
b
a
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
b
Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya
*Corresponding author: nhs@utm.my
1.0 INTRODUCTION
The commutativity degree is a concept that is used to determine
the abelianness of a group. It is denoted by  P G . The definition
of the commutativity degree is given as follows.
Definition 1.11
Let G be a finite non-abelian group. Suppose that
x and y are two random elements of .G The probability that two
random elements commute is given as follows.
 
  
2
, :
.
x y G G xy yx
P G
G
  

The first investigation of the commutativity degree of
symmetric groups was done in 1968 by Erdos and Turan2
. Few
years later, Gustafson3
and MacHale4
found an upper bound for
the commutativity degree of all finite non-abelian groups in which
 
5
.
8
P G  The concept of the commutativity degree has been
generalized and extended by several authors. In this paper, we use
one of these extensions, namely the probability that a group
element fixes a set. This probability was firstly introduced by
Omer et al5
in 2013.In this research, we apply it to the metacyclic
2-groups of positive type of nilpotency class two and class at least
three.
In 1975, a new concept was introduced by Sherman6
, namely
the probability of an automorphism of a finite group fixes an
arbitrary element in the group. The definition of this probability is
given as follows:
Definition 1.26
Let G be a group. Let X be a non-empty set of
G (G is a group of permutations of ).X Then the probability of
Article history
Received :24 April 2014
Received in revised form :
5 August 2014
Accepted :15 October 2014
Graphical abstract
 
  , | : ,
G
g s gs s g G s S
P S
S G
  

Abstract
In this paper, G is a metacyclic 2-group of positive type of nilpotency of class at least three. Let  be
the set of all subsets of all commuting elements ofG of size two in the form of  , ,a b where a and b
commute and each of order two. The probability that an element of a group fixes a set is considered as
one of the extensions of the commutativity degree that can be obtained by some group actions on a set.
In this paper, we compute the probability that an element of G fixes a set in which G acts on a set, 
by conjugation.
Keywords: Commutativity degree; metacyclic groups; group action
Abstrak
Dalam kertas kerja ini, G adalah satu kumpulan-2 metakitaran jenis positif dengan kelas nilpoten
sekurang-kurangnya tiga. Katalah  adalah set bagi semua subset untuk semua unsur kalis tukar tertib
bagi G dengan saiz dua dalam bentuk  ,a b di mana a dan b adalah kalis tukar tertib dan setiapnya
adalah berperingkat dua. Kebarangkalian bahawa suatu unsur dalam satu kumpulan menetapkan suatu
set dianggap sebagai salah satu daripada perluasan bagi darjah kekalisan tukar tertib yang boleh
didapati daripada beberapa tindakan kumpulan dalam suatu set. Dalam kertas kerja ini, kami mengira
kebarangkalian bahawa satu unsur bagi G menetapkan satu set yang mana G bertindak secara
kekonjugatan ke atas .
Kata kunci: Darjah kekalisan tukar tertib; kumpulan metakitaran; tindakan kumpulan
© 2014 Penerbit UTM Press. All rights reserved.

2. 8 Mustafa, Nor Haniza & Sanaa / Jurnal Teknologi (Sciences & Engineering) 71:1 (2014), 7–10
an automorphism of a group fixes a random element from X is
defined as follows:
 
   , , ,
.G
g x gx x g G x X
P X
G X
   

In 2011, Moghaddam7
explored Sherman's definition and
introduced a new probability, which is called the probability of an
automorphism fixes a subgroup element of a finite group, the
probability is stated as follows:
 
  , | : ,
, ,G
G
A
h h h H A
P H G
H G

  

where GA is the group of automorphisms of a group .G It is clear
that if ,H G then  , .G GA AP G G P
Next, we state some basic concepts that are needed in this paper.
Definition 1.38
A group G is called a metacyclic if it has a cyclic
normal subgroup H such that the quotient group G
H
is also
cyclic.
Definition 1.49
Let G be a finite group. Then, G acts on itself if
there is a functionG G G  , such that
   . , , , .
.1 , .G
i gh x g hx g h x G
ii x x x G
  
  
Definition 1.510
Let G be any finite group and X be a set. G acts
on X if there is a function ,G X X  such that
(1) ( ) ( ), , ,gh x g hx g h G x X   
(2)1 , .G x x x G  
Next, we provide some concepts related to metacyclic p-
groups. Throughout this paper, p denotes a prime number.
In 1973, King11
gave the presentation of metacyclic p-groups, as
1 2
1 2
given in the following:
, : 1, , , , where , 0,p p n m
G a b a b a b a a
 
         
 1 10, , | 1 .m n p p n m 
  
In 2005, Beuerle 12
separated the classification into two parts,
namely for the non-abelian metacyclic p-groups of class two and
class at least three. Based on12
, the metacyclic p- groups of
nilpotency class two are then partitioned into two families of non-
isomorphic p-groups stated as follows:
8.
, : 1, 1, , ,where , , ,
2 and 1.
p p p
G a b a b a b a
G Q
   
   
  

         
 

N
Meanwhile, the metacyclic p-groups of nilpotency class of at
least three (p is an odd prime) are partitioned into the following
groups:
, : 1, 1, , ,where , ,
, 1 2 and .
, : 1, , , ,where ,
, , , 1 2 , and .
p p p
p p p p
G a b a b b a a
G a b a b a b a a
   
     
  
   

          

 
       
   
       
     
N
N
Moreover, metacyclic p-groups are also classified into two
types, namely negative and positive12
. The following notations for
these two types which are used in this paper are represented as
follows:
 , , , , , : 1, , , ,
where , , , , 1.
p p p t
G a b a b a b a a
t p
   
 
   
   


        
  N
If 1t p 
  , then the group is called a metacyclic of negative
type and it is of positive type if 1t p 
  . Thus,
 , , , ,G     denotes the metacyclic group of negative type,
while  , , , ,G     denotes the positive type. These two
notations are shortened to  ,G p  and  ,G p  respectively11,12
.
The scope of this paper is only for the metacyclic p-groups of
positive type.
Theorem 1.112
Let G be a metacyclic 2-group of positive type of
nilpotency class of at least three. Then G is isomorphic to one of
the following types:
  2 2 2
1 , : 1, , , , , ,
1 2 .
G a b a b b a a
   
    
  

        
 


N
  2 2 2 2
2 , : 1, , , ,1 2 ,
, .
G a b a b a b a a
     
  
    
 
        
  

Definition 1.613
Let G act on a set S, and .x S If G acts on itself
by conjugation, the orbit O(x) is defined as follows:
   1
: for some .O x y G y axa a G
   
In this case O(x) is also called the conjugacy classes of x in
G. Throughout this paper, we use K(G) as a notation for the
number of conjugacy classes in G.
This paper is structured as follows: Section 1 provides some
fundamental concepts of group theory which are used in this
paper. In section 2, we state some of previous works, which are
related to the commutativity degree, in particular related to the
probability that a group element fixes a set. The main results are
presented in Section 3.
2.0 PRELIMINARIES
In this section, we provide some previous works related to the
commutativity degree, more precisely to the probability that an
element of a group fixes a set.

3. 9 Mustafa, Nor Haniza & Sanaa / Jurnal Teknologi (Sciences & Engineering) 71:1 (2014), 7–10
Recently, Omer et al.4
extended the commutativity degree by
defining the probability that an element of a group fixes a set of
size two, given in the following.
Definition 2.14
Let G be a group. Let S be a set of all subsets of
commuting elements of G of size two. If G acts on ,S then the
probability of an element of a group fixes a set is defined as
follows:
 
  , | : ,
.G
g s gs s g G s S
P S
S G
  

The following result from Omer et al.5
will be used in our
proof in the next section.
Theorem 2.14
Let G be a finite group and let X be a set of
elements of G of size two in the form of  ,a b where a and b
commute. Let S be the set of all subsets of commuting elements
of G of size two and G acts on S by conjugation. Then the
probability that an element of a group fixes a set is given by
 
 ,G
K S
P S
S
 where  K S is the number of conjugacy
classes of S in .G
Recently, Mustafa et al.14
has extended the work in 4
by
restricting the order of . The following theorem illustrates their
results.
Theorem 2.214
Let G be a finite group and let S be a set of
elements of G of size two in the form of  , ,a b where ,a b
commute and 2.a b  Let  be the set of all subsets of
commuting elements of G of size two and G acts on . Then
the probability that an element of a group fixes a set is given by
 
 ,G
K
P

 

where  K  is the number of conjugacy
classes of  in .G
3.0 MAIN RESULTS
This section provides our main results. First, we provide the
following theorem that illustrates the case when  GP  is equal to
one. Then, the probability that an element of metacyclic2-groups
of positive type of nilpotency class at least three fixes a set is
computed.
Theorem 3.1 Let G be a finite non-abelian group. Let S be a set
of elements of G of size two in the form of  ,a b where a and
b commute and 2.a b  Let  be the set of all subsets of
commuting elements of G of size two. If G acts on  by
conjugation, then   1GP   if and only if all commuting
elements a and b are in the center of .G
Proof First, suppose   1.GP   Then by Theorem 2.2,
  ,K    since G acts on  by conjugation, then there
  1
exists a function : such that , wheregG w gwg 
     
   , . Since , then the conjugacy class of ,w g G K a b    
     is cl , , , , .a b a b a b   By the conjugation thus
  1
cl , , .w gwg w g G
   Therefore, the conjugacy class of
      1 1
cl , , , , . Since thusa b gag gbg g G K 
     
       1 1
cl , , for all , . It follows that, ,a b a b a b gag gbg 
  
 ,a b thus 1 1
, .gag a gbg b 
  From which we have
,ga ag ,gb bg ,g G  . Since g and  ,a b are
arbitrary elements ofG and  respectively, then  , ,a b Z G as
claimed. Second, assume that  , .a b Z G Since the action here
is by conjugation, thus   1
cl , , .w gwg w g G
   Thus
   1 1
cl , , ,a b gag gbg 
 since  , ,a b Z G thus ,ag ga
, .bg gb g G   Therefore,  cl ,a b   , ,a b for all  , .a b 
Hence,   .K    Using Theorem 2.2,   1,GP   as
required.■
In the following two theorems, the probability that a group
element fixes a set of metacyclic 2-groups of positive type of
nilpotency class at least three is computed. In both theorems, let S
be a set of elements of G of size two in the form of (a, b), where a
and b commute and |a|=|b|=2. Let  be the set of all subsets of
commuting elements of G of size two and G acts on  by
conjugation.
Theorem 3.2 Let G be a finite group of type
  2 2 2
1 , , : 1, , ,G a b a b b a a
   
        where , , ,   N
1 2 , .         
2
Then .
| |
GP 


Proof IfG acts on  by conjugation, then there exists
:G  such that   1
, , .g w gwg w g G
    The
elements of order two in G are
1 1
2 2
,a b
  
and
1 1
2 2
.a b
  
Therefore, the elements of  are stated as follows: Two
elements in the form of  1 1 1
2 2 2
, ,0 2 ,i
a a b i
  
  
  and one
element in the form
1 12 1 2
2
, .b a b
   
 
 
Then 3.  If G acts on
 by conjugation then   1
cl , , .w gwg w g G
   The
conjugacy classes can be described as follows: One conjugacy
class in the form of  1 1 1
2 2 2
, ,0 2 ,i
a a b i
  
  
  and one
conjugacy class in the form of
1 12 1 2
2
, .b a b
   
 
 
Thus there are
two conjugacy classes. Using Theorem 2.2, then  
2
.GP  

■
Next, the probability that an element of a group of type (2)
fixes a set is computed.
Theorem 3.3 Let G be group of type   2
2 , , : 1,G a b a

  
2
b

 2 2
, ,a b a a
    
    ,1 2 , ,        .   
Then

4. 10 Mustafa, Nor Haniza & Sanaa / Jurnal Teknologi (Sciences & Engineering) 71:1 (2014), 7–10
 
2
, 1,
1, 1.
G
if
P
if
 
 

    
  
Proof First, let 1.   IfG acts on  by conjugation, then
  1
there exists : such that , ,gG w gwg w
      
4 1 1
7 2 2 2
. Thus, the elements of order two in are ,g G G a b
    
 

4 1
7 2 2
a b
  

and
1
2
.a

Hence, the elements of  are stated as
1 4 1 1
2 7 2 2 2
follows: Two elements are in the form of ( , ),i
a a b
      
 
0 1, and there is only one element in which is in the formi  
4 1 4 1 1
7 2 2 7 2 2 2
of ( , ),a b a b
        
  
from which it follows that
3.  Since the action is by conjugation, then the conjugacy
classes can be described as follows: One conjugacy class in the
form
1 4 1 1
2 7 2 2 2
( , ),0 1,i
a a b i
      
 
  and one conjugacy
class in the form
4 1 4 1 1
7 2 2 7 2 2 2
( , ).a b a b
        
  
Thus the
number of conjugacy classes is two. Using Theorem 2.2, then
 
2
.GP  

For the second case, namely when 1,   the
elements of order two in G are similar to the elements in the first
case. Hence, the elements of  are stated as follows: Two
elements in the form of
1 4 1 1
2 7 2 2 2
( , ),0 1,i
a a b i
      
 
  and
one element in the form
4 1 4 1 1
7 2 2 7 2 2 2
( , ).a b a b
        
  
Therefore, 3.  If G acts on  by conjugation, the
conjugacy classes can be described as follows: One conjugacy
class in the form
1 4 1
2 7 2 2
( , ),a a b
    

one conjugacy class in the
form of
1 4 1 1
2 7 2 2 2
( , )a a b
      
 
and one conjugacy class in the
form
4 1 4 1 1
7 2 2 7 2 2 2
( , ).a b a b
        
  
It follows that, there are
three conjugacy classes. Using Theorem 2.2, then   1.GP  
4.0 CONCLUSION
In this paper, the probability that a group element fixes a set is
found for metacyclic 2-groups of positive type of nilpotency class
at least three. Besides, the necessary and sufficient condition for
all non-abelian groups in which  GP  is equal to one under
conjugation action is provided.
Acknowledgement
The first author would like to acknowledge the Ministry of Higher
Education in Libya for the scholarship. The authors would also
like to acknowledge Ministry of Education (MOE) Malaysia and
Universiti Teknologi Malaysia for the financial funding through
the Research University Grant (GUP) Vote No 04H13.
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